The cross-catalytic CRN (REF ) exactly
recapitulates the large-deviation properties of the autocatalytic
CRN (REF ) along the axis of symmetry \(x_A =x_B\) , and approximates the solutions to the exact recursions, though
in more complicated form, at finite \(n_C\) , as derived
in [1]}. Its large-deviation behavior remains
simple in the sense that escape trajectories do not cross (so the
eikonal approximation is valid everywhere), and that they converge
strongly toward time-reverses of solutions to the mean-field equations
of motion (REF ) in neighborhoods of the
diagonal, leading to a computationally efficient approximation of the
effective potential \(\hat{\Xi } \! \left( x \right)\)  in that region.
Away from the diagonal, however, the irreversible nature of the CRN
becomes visible as escape trajectories deviate from mean-field
trajectories, leading to most-probable excursions that are loops, and
a non-trivial relation between the separatrices for mean-field flow
and for escapes. Here we provide details of these properties of the
eikonal approximation, retaining the same parameters \(\epsilon = 36\) ,
\(k_2 = 49\) , \(k_1 = 14\) , \({\bar{k}}_1 = 1\)  used for the 1-species CRN
from App. .
